\magnification = 2200 %\magstep3
%\vsize=1.05\vsize

\def\UseTimesRoman{
\font\cmr=Times
\font\TR=Times at 10pt
\font\TRXII=Times at 12pt
\font\TRXIV=Times at 14pt
\font\TRXX=Times at 20pt
\font\TRXXIV=Times at 24pt
\font\TI=TimesI at 10pt     %Times Italic
\font\TB=TimesB at 10pt     %Times Bold
\font\TBI=TimesBI at 10pt   %Times Bold Italic
\font\TBIviii=TimesBI at 8pt
\font\TBIv=TimesBI at 5pt
%\font\TO=TimesO at 10pt  %Times Oblique (Times Roman, slanted 22%
  %    with EdMetrics)
\font\TO=TimesI at 10pt  %Times Oblique (Times Roman, slanted 22% with EdMetrics)

\font\TIVIII=TimesI at 8pt
\font\TRVIII=Times at 8pt
\font\TIVI=TimesI at 6pt
\font\TRVI=Times at 6pt

	     \font\tenrmscld=Times at 10 pt
        \font\sevenrmscld=Times at 7 pt
        \font\fivermscld=Times at 5 pt

        \font\teniscld=cmmi10 at 10.3 pt
        \font\seveniscld=cmmi10 at 7.21 pt
        \font\fiveiscld=cmmi10 at 5.15 pt
        \font\tensyscld=cmsy10 at 10.3 pt
        \font\sevensyscld=cmsy10 at 7.21 pt
        \font\fivesyscld=cmsy10 at 5.15 pt
        \font\tenexscld=cmex10 at 10.3 pt
        \font\tenbfscld=cmbx10 at 10.3 pt
        \font\sevenbfscld=cmbx10 at 7.21 pt
        \font\fivebfscld=cmbx10 at 5.15 pt

\font\Courier = Courier
\font\Symbol = Symbol

\def\Omega{\hbox{{\Symbol W}}}

\textfont0=\tenrmscld \scriptfont0=\sevenrmscld\scriptscriptfont0=\fivermscld
\def\rm{\fam0\tenrmscld}
\textfont1=\teniscld \scriptfont1=\seveniscld \scriptscriptfont1=\fiveiscld
\def\mit{\fam1} \def\oldstyle{\fam1\teni}
\textfont2=\tensyscld \scriptfont2=\sevensyscld \scriptscriptfont2=\fivesyscld
\def\cal{\fam2}
\textfont3=\tenexscld \scriptfont3=\tenexscld \scriptscriptfont3=\tenexscld
\def\it{\TI}
\def\sl{\TO}
\def\bf{\TB}
\def\rm{\TR}
%\def\tt{\ttCourier}
\def\tt{\Courier}
\def\abstractfont{\TRVIII}
\def\footnotefont{\TRVIII}
\def\tinyfont{\TRvi}
\def\smalltitlefont{\TRXII}
\def\titlefont{\TRXIV}
\def\bigtitlefont{\TRXX}
\def\verybigtitlefont{\TRXXIV}
\textfont9=\TBI \scriptfont9=\TBIviii \scriptscriptfont9=\TBIv
\def\mbi{\fam9}
\rm
        }

%\UseTimesRoman

\def\BR{\Bbb R}             % Besondere Buchstaben
\def\BC{\Bbb C}
\def\BI{\Bbb I}
\def\BN{\Bbb N}
\def\BQ{\Bbb Q}
\def\BS{\Bbb S}
\def\BZ{\Bbb Z}
\def\Tilde{$_{\hbox{\cmrXX \~{}}}$}
\def\ST{\hbox{\eu T }}
\def\SRS{\hbox{\eu RS}}
\def\i{\hbox{{\bf i}}}

\font\sc=cmcsc10 at 10 pt   %% or: at 10 pt
\font\eu=eusb10 %at 10 pt
\font\small=cmr8 at 8 pt
\font\cmrX=cmbx10 scaled \magstep 1 %% 12 point CM
\font\cmrXX=cmbx12 scaled \magstep 1 %%

\hsize 7 true in
\vsize 9 true in
\hoffset = -0.20 true in
\voffset -0.25 true in
\parskip=3pt

\overfullrule = 0pt



\input amssym.def            % small letters for UNIX,  not: AMSsym.def
\input epsf.def% \input epsf %for UNIX
%\input epsf          %\input epsf.def for MAC f"ur BILDER!!
\input pics.tex

\input BoxedEPS
\SetTexturesEPSFSpecial
\HideDisplacementBoxes

\def\lf{\ \hfil\break}       % Neue Zeile ohne Einr"ucken, 'linefeed'
\def\cl{\centerline}
\def\Lf{\vskip1pt\noindent}   % Neue Zeile mit breiterem Zwischenraum
\def\LF{\medskip\noindent}   % Neue Zeile mit breiterem Zwischenraum
\def\R90{{\rm Rot}(90^\circ)}
\def\Dd#1{{\partial \over \partial #1}}

\nopagenumbers

\vglue -10pt
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\cl {\bf About  The Koch Snowflake }
\lf
\cl{ See also: Dragon Curve, Hilbert SquareFillCurve}
\cl { To speed up demos  press DELETE}
\Lf
  The Koch Snowflake Curve (aka the Koch Island) is 
a fractal planar curve of infinite length and Hausdorff
dimension approximately 1.262. It is defined as the limit 
of a sequence of polygonal curves defined recursively 
as follows:
\item{1)} The first polygon is an equilateral triangle.
\item{2)} The $(n + 1)$st polygon is created from the
    n-th polygon by applying the following rule to
    each edge:  construct an equilateral triangle 
    with base the middle third of the edge and 
    pointing towards the outside of the polygon,
    then remove the base of this new triangle.
  
\noindent
This is shown in the default demo of 3DXM.
  \Lf
At each step the number of segments increases by a 
factor 4 with the new segments being one third the 
length of the old ones. Since all end points of segments 
are already points on the limit curve we see that no part 
of the limit curve has finite length. 
\Lf
It is often illuminating to view a given Fractal curve as a
member of a continuous family of continuous curves of
increasing complexity. We generalize the Snowflake
construction, by changing the length ratio of those four smaller
segments that replace one segment of the previous
iteration. Instead of the length fraction one third we choose 
the length fraction $aa,\  0.25 \le  aa \le  0.5$, but we continue 
to arrange the two middle segments as an isocele triangle to 
the  left of the big segment. This family is the {\it default morph}.
It starts at $aa=1/4$ with an equilateral triangle, hits
the Snowflake at $aa=1/3$ and ends at $aa=1/2$ with
another continuous curve of positive area.
\Lf
The Dutch artist Escher created complicated tesselations by 
modifying each pair of neighboring tiles in such a way that they
still fit together. This can also be done with Fractal changes.
We use the following transition from the $n$-th to the $(n+1)$-st
iteration:
Replace each segment of the $n$-th polygonal approximation
by three segments of equal length in such a way that the midpoint 
of the second segment is also the midpoint of the previous
segment, the vertex between the first and second is always to
the left and the vertex between the second and the third segment
is always to the right of the previous segment. This is illustrated
if one selects in the
\lf
Action Menu: {\it Choose Escher Version.}
\lf
Note that for varying length of the new (short) segments the two
new vertices lie on a circle that is independent of the new length.
This Escher family is therefore parametrized by the angle $bb$ at
the midpoint of this circle, $0 \le bb \le \pi/3$. The demo shows two
adjacent copies of the approximations to illustrate that they indeed fit 
together because of the point symmetry of each modification with 
respect to the midpoint. Pairs of limit curves also fit together.
The {\it default morph} starts again from an equilateral triangle 
(two copies) and ends at another continuous curve of positive area. 
The  number of computed iterations is controlled with $ee$.
\Lf 
\vskip2mm

\cl{\bf On the Hausdorff Dimension of the Snowflake}
\vskip2mm \noindent
 Consider the union of those disks that have a segment 
of one polygonal approximation as a diameter, then this 
union covers all the further approximations. From one step 
to the next the diameter of the disks shrinks to one third
while the number of disks is multiplied by 4---so that the 
area of these covering disk unions converges to zero. 
The fractal Hausdorff \hbox{d-measure} is defined as the infimum
(as the diameter goes to zero) of:
                       $$(diameter^d \cdot  number\ of\ disks)$$
and the fractal Hausdorff dimension is the infimum of 
those $d$ for which this $d$-measure is 0. This shows that the 
Hausdorff dimension of the Koch curve is less than or equal to   
$\log(4) / \log(3)$, and since the union of the disks of every second 
segment does not cover the limit curve one can conclude that 
the Hausdorff dimension is precisely $\log(4) / \log(3)$.


\bye
 

 